In the design of singlet microwave dielectric lenses in the electromagnetic spectrum and acoustic lenses for underwater sound applications or for certain applications in the optical spectrum in which wide angle scanning or fixed off-axis operation is required, it is frequently desired to achieve greater off-axis scan in one plane than in the other, wide angle azimuth performance usually being more necessary than elevation scan. Due to requirements for weight minimization and compact packaging for aircraft and submersible vehicle installations where both system volume and weight are at a premium, it is further generally desirable to extremize the lens in the sense of maximizing the lens aperture or minimizing the lens volume and sometimes simultaneously to achieve a minimum, or at any rate a near minimum, F-number.
On the other hand, in both the microwave and acoustic applications for the most part, excluding only those areas of medical acoustic radiography and related fields in which frequencies higher than 10 megahertz are commonly used, the wavelengths encountered are of the order of a millimeter, a centimeter, or a decimeter so that there is no need to restrict the lens design to spherical surfaces grindable only by self-correcting motions as in optical lens manufacture, but rather, nonspherical lenses of quite complex shape may be readily used. With this relaxation of the design requirements, consideration in radar, and more recently in acoustics, was first given to nonspherical but rotationally invariant lenses, i.e., nonspherical lenses whose surfaces are surfaces of revolution about the lens axis such as the aplanatic, generalized aplanatic and bifocal or Doppler lenses of L. C. Martin, F. G. Friedlander and R. L. Sternberg. Such lenses, however, like any ordinary singlet lens, spherical or nonspherical, invariably suffer when extremized from astigmatism at off-axis points with the magnitude of that aberration increasing with increasing scan angle and decreasing F-number.